It is important to highlight here that exchangeability is one of the four basic probabilistic symmetries defined by [1]}; namely, stationarity, contractability, exchangeability, and rotatability. These four distributional symmetries correspond to invariance properties under the four respective transformations of shifts, sub-sequences, permutations, and isometries. The four probabilistic symmetries form a hierarchy with stationarity at the bottom and rotatability, the strongest invariance property, at the top. Exchangeability in particular allows us, through conditioning, to transform the joint density into a mixture of densities.
